Text of Optimum Image Formation for Spaceborne Microwave...
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016 2763
Optimum Image Formation for SpaceborneMicrowave Radiometer Products
David G. Long, Fellow, IEEE, and Mary J. Brodzik
Abstract—This paper considers some of the issues of radiometerbrightness image formation and reconstruction for use in theNASA-sponsored Calibrated Passive Microwave Daily Equal-AreaScalable Earth Grid 2.0 Brightness Temperature Earth SystemData Record project, which generates a multisensor multidecadaltime series of high-resolution radiometer products designed tosupport climate studies. Two primary reconstruction algorithmsare considered: the Backus–Gilbert approach and the radiometerform of the scatterometer image reconstruction (SIR) algorithm.These are compared with the conventional drop-in-the-bucket(DIB) gridded image formation approach. Tradeoff study resultsfor the various algorithm options are presented to select optimumvalues for the grid resolution, the number of SIR iterations, andthe BG gamma parameter. We find that although both approachesare effective in improving the spatial resolution of the surfacebrightness temperature estimates compared to DIB, SIR requiressignificantly less computation. The sensitivity of the reconstructionto the accuracy of the measurement spatial response function(MRF) is explored. The partial reconstruction of the methods cantolerate errors in the description of the sensor measurement re-sponse function, which simplifies the processing of historic sensordata for which the MRF is not known as well as modern sensors.Simulation tradeoff results are confirmed using actual data.
Index Terms—Brightness temperature, climate data record,Earth system data record, passive microwave remote sensing,radiometer, reconstruction, sampling, variable aperture.
SATELLITE passive microwave observations of surfacebrightness temperature are critical to describing and under-
standing Earth system hydrologic and cryospheric parametersthat include precipitation, soil moisture, surface water, vege-tation, snow water equivalent, sea-ice concentration, and sea-ice motion. These observations are available in two forms:swath (sometimes referred to as NASA EOSDIS Level 1A orLevel 1B) data and gridded (Level 3) data . While swathdata have applications for processes with short timescales,gridded data are valuable to researchers interested in derivedparameters at fixed locations through time and are widely
Manuscript received September 14, 2015; revised October 31, 2015 andNovember 27, 2015; accepted November 30, 2015. Date of publicationDecember 25, 2015; date of current version March 25, 2016.
D. G. Long is with the Department of Electrical and Computer Engineering,Brigham Young University, Provo, UT 84602 USA (e-mail: [email protected]).
M. J. Brodzik is with the Cooperative Institute for Research in EnvironmentalSciences, University of Colorado Boulder, Boulder, CO 80309-0216 USA, andalso with the National Snow and Ice Data Center, University of ColoradoBoulder, Boulder, CO 80309-0449 USA (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TGRS.2015.2505677
used in climate studies. Both swath and gridded data fromthe current time series of satellite passive microwave datasets span over 30 years of Earth observations, beginning withthe Nimbus-7 Scanning Multi-channel Microwave Radiometer(SMMR) sensor in 1978, continuing with the Special SensorMicrowave/Imager (SSM/I) and Special Sensor MicrowaveImager/Sounder (SSMIS) series sensors from 1987 onward,and including the completed observational record of AquaAdvanced Microwave Scanning Radiometer-Earth ObservingSystem (AMSR-E) from 2002 to 2011. Although there arevariations between sensors, this data record is an invaluableasset for studies of climate and climate change.
A number of heritage gridded products are available, includ-ing those from the National Snow and Ice Data Center (NSIDC)(see –). Although widely used in a variety of scientificstudies, the processing and spatial resolution of existing griddeddata sets are inconsistent, which complicates long-term climatestudies using them. Each of the currently available gridded datasets suffers from inadequacies as Earth System Data Records[ESDRs, also referred to as Climate Data Records (CDRs)]since most of them were developed prior to the establishment offormal definitions for ESDR/CDRs. Perhaps, the most criticallimitation of heritage products is the lack of cross-calibration ofthe sensors. In addition, all of the existing gridded data sets atNSIDC employ the original EASE-Grid projection , whichhas since been revised to be used more easily with standardgeospatial mapping programs , . Furthermore, since defi-nition of these heritage products, new image reconstruction andinterpolation schemes have been developed, which can be usedto improve the products.
The SSM/I-SSMIS swath data record has been recentlyreprocessed and cross-calibrated by two research teams whohave published the data as fully vetted fundamental CDRs(FCDRs): Remote Sensing Systems (Santa Rosa, CA) and Colorado State University . To exploit the availabilityof these new FCDRs and improved gridding schema whileameliorating the limitations of current gridded data sets, theNASA MEaSUREs Calibrated Passive Microwave Daily Equal-Area Scalable Earth (EASE) Grid 2.0 Brightness Temperature(CETB) Earth System Data Record (ESDR)  is gener-ating a single consistently processed multisensor ESDR ofEarth-gridded microwave brightness temperature (TB) images.The multi-decadal product includes sensor data from SMMR,SSM/I, SSMIS, and AMSR-E with all the improved swathdata sensor calibrations recently developed, as well as improve-ments in cross-sensor calibration and quality checking, modernfile formats, better quality control, improved grid projection
2764 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
definitions , , improved gridding techniques, and local-time-of-day (ltod) processing. To exploit developments inresolution enhancement, the CETB ESDR includes both con-ventional and enhanced resolution products. Designed to servethe land surface and polar snow/ice communities, the newproducts are intended to replace existing heritage griddedsatellite passive microwave products with a single consistentlyprocessed ESDR .
A key part of processing is conversion of the swath-basedmeasurements to the Level 3 grid. Algorithms to transformradiometer data from swath to gridded format are character-ized by a tradeoff between noise and spatial and temporalresolution. Conventional drop-in-the-bucket (DIB) techniquesprovide low-noise low-resolution products, but higher resolu-tion (with potentially higher noise) products are possible usingimage reconstruction techniques. By providing products withboth processing options, users can compare and choose whichoption better suits their particular research application. Thepurpose of this paper is to explore the options for the high-resolution processing products and to compare the results toconventional resolution processing. A key goal of this paper isto describe the methods and their tradeoffs.
This paper is organized as follows. After some brief back-ground, a review of the theory of radiometer image reconstruc-tion is provided that includes a derivation of the radiometermeasurement spatial response function (MRF) and a discussionof radiometer image formation algorithms. The succeedingsection employs simulation to select the optimum parametersfor image formation for the SSM/I sensor and to evaluate thesensitivity of the reconstruction to the accuracy of the responsefunction. Actual data results are provided in the followingsection, followed by a summary conclusion. An Appendixconsiders the spatial frequency response of a radiometer MRF.
Previous gridded passive microwave data sets have used var-ious swath-to-grid interpolation schemes on different grids fordifferent radiometer channels. Some products  used classicDIB methods described in Section IV-A, whereas others , used inverse-distance squared weighting. These approachesresulted in low-resolution gridded data with low noise, at theexpense of spatial resolution. However, there is interest in theuser community for higher resolution products.
For the conventional-resolution CETB product, all radiome-ter channels are gridded to a single coarse resolution gridusing DIB method described in Section IV-A. To also pro-vide a high spatial resolution product, the CETB exploits theresults of previous studies that have demonstrated that high-resolution radiometer images can be produced using processingalgorithms based on the Backus–Gilbert (BG) ,  ap-proach – and the radiometer form of the scatterometerimage reconstruction (SIR) algorithm , , which hasbeen successfully applied to SMMR, SSM/I, and AMSR-Edata , .
BG was employed in generating the swath-based griddedSSM/I Pathfinder EASE-Grid product . The BG implemen-tation used data from a single swath even when overlapping
swaths were available. Using a single swath reduces processingrequirements by enabling precalculation of BG weights versusscan position, with tuning parameters set for low noise (andtherefore low spatial resolution). The antenna footprints wereassumed circular. Gridded data were produced separately forascending and descending passes.
The Brigham Young University Scatterometer ClimatePathfinder (SCP, www.scp.byu.edu) generated an enhanced-resolution passive microwave data set for selected periods ofSSM/I and AMSR-E. Available from NSIDC , the SCPproduct uses the SIR algorithm to report TB on finer spatialgrids than possible with conventional processing. This productcombines data from multiple passes using measurement ltod,which minimizes fluctuations in the observed TB at high lati-tudes due to changes in physical temperature from daily tem-perature cycling. Two images per day are produced, separatedby 12 hours (morning and evening), with improved temporalresolution, permitting resolution of diurnal variations . Theenhanced resolution gridded data are proving useful in a varietyof scientific studies, e.g., in  and .
Both BG and SIR use regularization to tradeoff noise andresolution. While BG is based on least squares and depends ona subjectively chosen tradeoff parameter for regularization, SIRemploys maximum entropy reconstruction with regularizationaccomplished by limiting the number of iterations and therebyonly producing partial reconstruction. BG and SIR providesimilar results, although SIR offers a computational advantageover BG . Based on ongoing feedback from an early adoptercommunity, the final CETB product will likely contain imagesderived from only one of the candidate image reconstructionmethods. To inform this decision, one of the goals of this paperis to describe the methods and their tradeoffs.
The passive microwave sensors employed in the CETB flyon near-polar sun-synchronous satellites, which maintain anorbital plane with an orientation that is (approximately) fixedwith respect to the sun. Thus, the satellite crosses the equatoron its ascending (northbound) path at the same ltod (withinsmall tolerance). The resulting coverage pattern yields passesabout 12 h apart in ltod at the equator. Most areas near the poleare covered multiple times per day. Analyzing the data from asingle sensor, we find that polar measurements fall into two ltodranges. The two periods are typically less than 4 h long and arespaced 8 or 12 h apart. Significantly, at any particular location,the orbital pass geometry causes the ltod of the observationsto differ from day to day in a cyclic manner . When notproperly accounted for, this can introduce undesired variability(noise) into a time series analysis due to diurnal variations ofthe surface that is sampled at different ltod over the multidayorbit repeat cycle.
Heritage gridded TB products have either 1) averaged allmeasurements during the day that fall in a given grid cell ;or 2) selected measurements from only one (ascending or de-scending) pass per day –. The observed microwave bright-ness temperature is the product of surface physical temperatureand surface emissivity. Since surface temperatures can fluctuate
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Fig. 1. Illustrations of the boundaries along the swath for (left) ascending/descending and (right) ltod for a northern hemisphere swath. The ascending/descending division is based on the nadir position of the spacecraft and, dueto the antenna rotation, results in significant ltod spread between passes atthe poles. The ltod approach segments the swath based on the ltod of eachmeasurement and results in more consistent swath division.
widely during the day, daily averaging (method 1) is not gen-erally useful since it smears diurnal temperature fluctuations inthe averaged TB . The single-pass approach (method 2) discardslarge amounts of potentially useful data. This method separatesdata into ascending pass-only and descending pass-only data,resulting in two images per day. This is a reasonable approach atlow latitudes, but at higher latitudes, the ascending/descendingdivision does not work as well since adjacent pixels along swathoverlap edges can come from widely different ltod (which varyon subsequent days) . The gridded field of TB’s, ostensiblyall representing consistent ltods, actually represent differentphysical temperature conditions. Thus, some modified form ofswath averaging is desired.
Another alternative is to split the data into two images perday, based on the ltod approach of Gunn and Long .Fig. 1 shows the difference in where the ascending/descendingdivision and ltod divisions occur. Particularly near the pole, theltod approach ensures that all measurements in any one im-age have consistent spatial/temporal relationships. The CETBadopts the ltod division scheme for the northern and southernhemispheres. At low latitudes, (which typically have few over-lapping swaths at similar ltod in the same day), the ltod divisionis equivalent to the ascending/descending division. For theCETB product, an ancillary image is included to describe theeffective time average of the measurements combined intothe pixel for a particular day. This enables investigators toexplicitly account for the ltod temporal variation of the mea-surements included in a particular pixel. To account for thedifferences in orbits of the different sensors, the precise ltoddivision time for the twice-daily images varies among sensors.
III. RADIOMETER SPATIAL RESPONSE FUNCTION
The effective spatial resolution of gridded image productsis determined by the MRF of the sensor and by the imageformation algorithm used. The MRF is determined by theantenna gain pattern, which is unique for each sensor and sensorchannel, and varies with scan angle, the scan geometry (notablythe antenna scan angle), and the measurement integration pe-riod. This section derives the MRF for a microwave radiometer.First, basic background is provided, followed by a discussionof the effects of temporal integration.
A. Background Theory
Microwave radiometers measure the thermal emission,which is sometimes called Plank radiation, radiating fromnatural objects . In a typical radiometer, an antenna isscanned over the scene of interest, and the output power fromthe carefully calibrated receiver is measured as a function ofscan position. The reported signal is a temporal average of thefiltered received signal power .
The observed power is related to receiver gain and noise fig-ure, antenna loss, physical temperature of the antenna, antennapattern, and scene brightness temperature. In simplified form,the output power PSYS of the receiver can be written as 
PSYS = kTSYSB (1)
where k = 1.38× 10−23 (J/K) is Boltzmann’s constant, Bis the receiver bandwidth in hertz, and TSYS is the systemtemperature in Kelvin with
TSYS = ηlTA + (1 − ηl)Tp + (L− 1)Tp + LTREC (2)
where ηl is the antenna loss efficiency (unitless), Tp is thephysical temperature in Kelvin of the antenna and waveguidefeed, L is the waveguide loss (unitless), TREC is the effectivereceiver noise temperature (determined by system calibration)in Kelvin, and TA is the effective antenna temperature inKelvin. As described in the following, the effective antennatemperature is dependent on the direction the antenna pointsand on the scene characteristics. Since the other instrument-related terms (i.e., (1− ηl)Tp + (L− 1)Tp + LTREC) can beremoved by proper calibration, the value of TA, which dependson the geophysical parameters of interest, is thus estimatedfrom TSYS.
The effective antenna temperature TA can be modeled as aproduct of the apparent temperature distribution TAP(θ, φ) inthe look direction θ, φ, and the antenna radiation gain F (θ, φ),which is proportional to the antenna gain pattern G(θ, φ) .TA (in Kelvin) is obtained by integrating the product of ap-parent temperature distribution TAP(θ, φ) (in Kelvin) and theunitless antenna pattern G(θ, φ):
TA = G−1a
∫ ∫G(θ, φ)TAP(θ, φ)dθdφ (3)
∫ ∫G(θ, φ)dθdφ (4)
and G(θ, φ) is the instantaneous antenna gain for the particularchannel and where the integrals are over the range of valuescorresponding to the nonnegligible gain of the antenna. Notethat the antenna pattern acts as a nonideal low-pass spatialfilter of the surface brightness distribution, limiting the primarysurface contribution to the observed TB to approximately 3-dBbeamwidth, although the observed value includes contributionsfrom a larger area.
For downward-looking radiometers, the apparent brightnesstemperature distribution includes contributions from the surface
2766 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
and from the intervening atmosphere . For a spacebornesensor, this can be expressed as
where TB(θ, φ) is the surface emission brightness tempera-ture, Tss(θ, φ) is the surface scattered radiation, τ is the totaleffective optical depth of the atmosphere, and Tup(θ) is theeffective atmospheric upwelling temperature, which dependson the temperature and density profile, atmospheric losses,clouds, rain, etc.
Ignoring incidence and azimuth angle dependence, the sur-face brightness temperature is
TB = εTP (6)
where ε is the emissivity of the surface, and TP is the physicaltemperature of the surface. The emissivity is a function ofthe surface roughness and permittivity, which are related tothe geophysical properties of the surface . In geophysicalstudies, the ultimate key parameter of interest is typically ε,although TP is also of interest.
The surface scattering temperature Tss(θ, φ) is the result ofdownwelling atmospheric emissions that are scattered off ofthe rough surface toward the sensor. This signal depends onthe scattering properties of the surface (surface roughness anddielectric constant) as well as the atmospheric emissions di-rected toward the ground. We note that, although some azimuthvariation with brightness temperature has been observed overthe ocean , sand dunes , and snow in Antarctica ,vegetated and sea-ice-covered areas generally have little or noazimuth brightness variation .
B. Signal Integration
The received radiometer signal is very noisy. To reduce themeasurement variance, the received signal power is averagedover a short integration period. Even so, the reported measure-ments are noisy due to the limited integration time available foreach measurement. The uncertainty is expressed as ΔT , whichis the standard deviation of the temperature measurement. ΔTis a function of the integration time and bandwidth used tomake the radiometric measurement and is typically inverselyrelated to the time–bandwidth product . Increasing the inte-gration time and/or bandwidth reduces ΔT . High stability andprecise calibration of the system gain are required to accuratelyinfer the brightness temperature TB from the sensor powermeasurement PSYS.
Because the antenna is scanning during the integration pe-riod, the effective antenna gain pattern of the measurements isa smeared version of the antenna pattern. In the smeared case,we replace G in (3) and (4) with the smeared version of theantenna, Gs, where
Gs(θ, φ) = T−1i
∫G(θ, φ + ωrt)dt (7)
with Ti being the integration period, ωr being the antennarotation rate, and the integral limits being 0 and Ti. Note thatbecause Ti is very short, the net effect is primarily to widen
the main lobe. Nulls in the pattern tend to be eliminated andthe sidelobes widened. The smeared antenna pattern variessomewhat for different antenna azimuth angles, although forsimplicity, this effect is not considered in this paper.
For imaging the surface, we can concentrate on the patternsmearing at the surface. The smeared antenna pattern Gs(θ, φ)at the surface at a particular time defines the MRF of thecorresponding TB measurement. Due to varying geometry, theMRF of different measurements may be slightly different. Forexample, the azimuth angle changes across the swath result invarying rotation angle of the MRF.
Note from (5) that TAP(θ, φ) consists primarily of an atten-uated contribution from the surface (i.e., TB) plus scatteredand upwelling terms. We note that the reported TA valuescompensate or correct (to some degree) for these terms.
Let T ′A denote the corrected TA measurement. It follows that
we can rewrite (5) in terms of the corrected TA and the surfaceTB value as
T ′A = G−1
∫ ∫Gs(θ, φ)TB(θ, φ)dθdφ. (8)
We can express this result in terms of the surface coordinatesx and y by noting that, for a given x, y location and time,the antenna elevation and azimuth angles can be computed(see the Appendix). Then
T ′A = G−1
∫ ∫Gs(x, y)TB(x, y)dxdy (9)
∫ ∫Gs(x, y)dxdy. (10)
In surface coordinates, the MRF is defined as
MRF(x, y) = G−1b Gs(x, y) (11)
T ′A =
∫ ∫MRF(x, y)TB(x, y)dxdy. (12)
Thus, the measurements T ′A can be seen to be the integral
of the product of the MRF and the surface brightness tem-perature. The nominal resolution of the TB measurements isgenerally considered the size of the 3-dB response pattern of thesmeared MRF. Image formation estimates TB(x, y) from themeasurements T ′
IV. GRIDDING AND RECONSTRUCTION
Algorithms that generate 2-D gridded images from rawmeasurements are characterized by a tradeoff between noiseand spatial resolution. Our goal is to estimate an image ofthe surface TB(x, y) from the sensor TB measurements. The“nominal” resolution of the TB measurements is typicallyconsidered the size of the 3-dB response pattern of the MRF.Although the effective resolution of DIB imaging is no finer
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than the effective resolution of the measurements, reconstruc-tion techniques can yield higher effective resolution if spatialsampling requirements are met.
As previously noted, the CETB project  is to produceboth low-noise gridded data and enhanced-resolution data prod-ucts. The low-resolution gridded data uses the DIB methoddescribed in the following. These products are termed “lowresolution” or “nonenhanced resolution” and are denoted asgridded (GRD) products. Higher resolution products are alsogenerated using one of the two image reconstruction methods:the radiometer form of the SIR algorithm or the BG imageformation method, as described in the following. Unlike theapproach taken for the historical EASE-Grid data , theCETB does not attempt to degrade the resolution of the highestchannels to match the coarsest channel; rather, it independentlyoptimizes the resolution for each channel in the high-resolutionproducts. The product is Earth-located (in contrast to swathbased) using EASE-Grid 2.0 , . In generating CETB grid-ded data, only measurements from a single sensor and a channelare processed. Measurements combined into a single grid el-ement may have different Earth azimuth and incidence angles(although the incidence angle variation is small). Measurementsfrom multiple orbit passes over a narrow local time windowmay be combined. When multiple measurements are com-bined, the resulting images represent a temporal average of themeasurements over the averaging period. There is an implicitassumption that the surface characteristics remain constant overthe imaging period and that there is no azimuth variation inthe true surface TB . For both conventional (nonenhanced) res-olution and enhanced-resolution images, the effective griddedimage resolution depends on the number of measurements andthe precise details of their overlap, orientation, and spatiallocations.
The succeeding sections provide a brief summary of thealgorithms used for image formation. Channels are gridded atenhanced resolution on nested grids at power-of-two relation-ships to the 25-km base grid. This embedded gridding simplifiesoverlaying grids from different resolutions. The method fordetermining the fine-resolution grid for each channel is givenin the following.
A. Coarse-Resolution GRD Gridding Algorithm
A classic coarse-resolution gridding procedure is the simpleDIB average. The resulting data grids are designated GRD dataarrays. For the DIB gridding algorithm, the key informationrequired is the location of the measurement. The center ofeach measurement location is mapped to an output projectedgrid cell. All measurements within the specified time periodwhose center locations fall within the bounds of a particulargrid cell are averaged together. The unweighted average be-comes the reported pixel value. Although some investigators,  have interpolated measurement locations to the center,or weighted the measurements by the distance to the center,such interpolation increases the noise level; therefore, in theCETB no interpolation or measurement weighting is done.CETB ancillary data contain the number and standard deviationof included samples.
Fig. 2. GRD pixel resolution illustration. The inner square represents the pixelarea while the 3-dB footprints of some potential measurements, whose centeris shown as a dot, are placed within the pixel. Note that more than half of somemeasurement can extend outside the pixel area. Depending on the measurementdistribution, to the first order, the dimensions of the effective pixel size are thusthe sum of the pixel and the footprint dimensions as indicated by the dashedbounding box.
The effective spatial resolution of the GRD product is definedby a combination of the pixel size and spatial extent of the 3-dBantenna footprint size ,  and does not require anyinformation about the antenna pattern. Although the pixel sizecan be arbitrarily set, the effective resolution is, to the firstorder, the sum of the pixel size plus the footprint dimension(see Fig. 2). All CETB GRD products are produced on a 25-kmpixel grid and thus have an effective resolution that is coarserthan 25 km since the measurement footprints can extend outsideof a pixel.
B. Reconstruction Algorithms
In the reconstruction algorithms, the effective MRF for eachmeasurement is used to estimate the surface TB on a fine-scale grid. The MRF is determined by the antenna gain pattern(which is unique for each sensor and sensor channel, and varieswith scan angle), the scan geometry (notably the antenna scanangle), and the integration period. The latter “smears” the an-tenna gain pattern due to antenna rotation over the measurementintegration period. The MRF describes how much the emissionsfrom a particular receive direction contribute to the observedTB value.
Denote the MRF for a particular channel by R(φ, θ;φi),where φ and θ are particular azimuth and elevation anglesrelative to the antenna boresite at azimuth scan angle φi. Notethat, for a given antenna azimuth scan angle, the MRF isnormalized so that the integral of the MRF over all azimuthand elevation angles is one.
Generally, for the FCDR data sets that are input to the CETB,the MRF can be treated as zero everywhere but in the directionof the surface. With this assumption, we can write R(φ, θ;φi)as R(x, y;φi), where x and y are the location (which we willexpress in map coordinates) on the surface corresponding tothe azimuth and elevation angles for the particular φi. Note that∫ ∫
R(x, y;φi)dxdy = 1. A particular noise-free measurementTi taken at a particular azimuth angle φi can be written as
∫ ∫R(x, y;φi)TB(x, y;φi)dxdy (13)
2768 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
where TB(x, y;φi) is the nominal brightness temperature inthe direction of point x, y on the surface as observed from thescan angle φi. Note that, if there is no significant differencein the atmospheric contribution as seen from different scanangles and if the surface brightness temperature is azimuthallyisotropic, we can treat TB(x, y;φi) as independent of φi so thatTB(x, y;φi) = TB(x, y). TB(x, y) is referred to as the surfacebrightness temperature.
With this approximation, a typical brightness temperaturemeasurement can be written as
∫ ∫R(x, y;φi)TB(x, y)dxdy + noise (14)
where noise is radiometric measurement noise. Each measure-ment is seen to be the MRF-weighted average of TB. The goalof the reconstruction algorithm is to estimate TB(x, y) from themeasurements Ti.
For the problem at hand, TB is desired at each sample pointon the EASE-Grid 2.0 grid. While this is a regular grid, themeasurement locations are not aligned with the grid, and themeasurements form an irregular sampling pattern. There is awell-defined theory of signal reconstruction based on irregularsampling that can be applied to the problem , , .However, since the signal measurements are quite noisy, fullreconstruction can produce excessive noise enhancement. Toreduce noise enhancement and resulting artifacts, at the expenseof resolution, regularization can be employed . Regulariza-tion is a smoothing constraint introduced to an inverse problemto prevent extreme values or overfitting.
The regularization in BG and SIR enable a tradeoff betweensignal reconstruction accuracy and noise enhancement. SIR isbased on signal processing and treats the surface brightnesstemperature as a 2-D signal to be reconstructed from irregularsamples (the measurements). BG is a least-squares approachthat trades noise and solution smoothness using a subjec-tively selected parameter , . While related alternateapproaches exist , , , they are not considered heredue to limited validation for this application and because theirresults are generally similar.
Both the BG and SIR approaches enable estimation of thesurface brightness on a finer grid than is possible with theconventional DIB approaches, i.e., the resulting brightnesstemperature estimate has a finer effective spatial resolutionthan DIB methods. As a result, the results are often called“enhanced resolution,” although in fact, the reconstruction al-gorithm merely exploits the available information to reconstructthe original signal at higher resolution than GRD gridding,based on the assumption of a bandlimited signal . Com-pared with the GRD coarsely gridded product, the potentialresolution enhancement depends on the sampling density andthe MRF; however, improvements of 25% to 1000% in theeffective resolution have been demonstrated in practice for par-ticular applications. For radiometer enhancement, the effectiveimprovement in resolution tends to be limited and in practiceis typically less than 100% improvement. Nevertheless, theresulting images have improved spatial resolution and infor-mation. Note that to meet Nyquist requirements for the signal
processing, the pixel resolution of the images must be finer thanthe effective resolution by at least a factor of two.
For comparison, note that the effective resolution for DIBgridding is equivalent to the sum of pixel grid size plus the spa-tial dimension of the measurement, which is typically definedby the half-power or the 3 dB beamwidth. Based on Nyquistconsiderations, the highest representable spatial frequency forDIB gridding is twice the grid spacing.
Note that, in some cases (notably in the polar regions),multiple passes over the same area at the same ltod can beaveraged together. Reconstruction algorithms intrinsically ex-ploit the resulting oversampling of the surface to improve theeffective spatial resolution in the final image.
C. Signal Reconstruction and SIR
In the reconstruction, TB(x, y) is treated as a discrete sig-nal sampled at the map pixel spacing and is estimated fromthe noisy measurements Ti. The pixel spacing must be setsufficiently fine so that the generalized sampling requirements are met for the signal and the measurements , .Typically, this means that the pixel spacing must be one-half toone-tenth the size of antenna footprint size.
To briefly describe the reconstruction theory, for conve-nience, we vectorize the 2-D signal over an Nx ×Ny pixelgrid into a single-dimensional variable aj = TB(xj , yj), wherej = l +Nxk. The measurement equation (14) becomes
where hij = R(xl, yk;φi) is the discretely sampled MRF forthe ith measurement evaluated at the jth pixel center and thesummation is over the image with
∑j hij = 1. In practice, the
MRF is negligible at some distance from the measurement;therefore, the sums need only to be computed over an area localto the measurement position. Some care has to be taken nearimage boundaries.
For the collection of available measurements, (15) can bewritten as the matrix equation, i.e.,
�T = H�a (16)
where H contains the sampled MRF for each measurement, and�T and �a are vectors composed of the measurements Ti and thesampled surface brightness temperature aj , respectively. Notethat H is very large, sparse, and may be overdetermined orunderdetermined depending on the sampling density. Estimat-ing the brightness temperature at high resolution is equivalentto inverting (16). In the case of the CETB, H is typicallyunderdetermined, and if explicitly written out, would havedimensions of over 106 × 106.
Due to the large size of H, iterative methods are the mostpractical approach to inverting (16). One advantage of an iter-ative method is that regularization can be easily implementedby prematurely terminating the iteration; alternately, explicitregularization methods such as Tikhonov regularization can be used.
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The radiometer form of SIR is a particular implementationof an iterative solution to (16) that has proven effective ingenerating high-resolution brightness temperature images .The SIR estimate approximates a maximum-entropy solutionto an underdetermined equation and a least squares solution toan overdetermined system. The first iteration of SIR is termed“AVE” (for weighted average) and can be a useful estimate ofthe surface TB. The AVE estimate of the jth pixel is given by
∑i hijTi∑i hij
where the sums are over all measurements that have nonnegli-gible MRF at the pixel.
D. BG Method
A direct approach to inverting (16) is based on Backus andGilbert ,  who developed a general method for invert-ing integral equations. The BG method is useful for solvingsampled signal reconstruction problems . First applied tothe radiometer data in , the BG method has been usedextensively for extracting vertical temperature profiles fromradiometer data . It has also been used for spatially inter-polating and smoothing data to match the resolution betweendifferent channels  and to improve the spatial resolution ofsurface brightness temperature fields , , . Antennapattern deconvolution approaches have also been attemptedwith varying degrees of success , .
In application to image reconstruction of the surface bright-ness temperature, the essential idea is to write an estimate ofthe surface brightness temperature at a particular pixel as aweighted linear sum of measurements that are collected “close”to the pixel. Using the notation developed earlier, the estimateat the jth pixel is
where the sum is computed over nearby pixels and where wij
are weights selected so that∑
iwij = 1.There is no unique solution for the weights; however, regular-
ization permits a subjective tradeoff between the noise level inthe image and the resolution . Regularization and selectionof the tuning parameters are described in detail elsewhere ,. There are two tuning parameters: an arbitrary-dimensionalparameter and a noise-tuning parameter γ. The dimensionalparameter affects the optimum value of tuning parameter γ.Following Robinson et al. , we set the dimensional-tuningparameter to 0.001. The noise-tuning parameter, which can varyfrom 0 to π/2, controls the tradeoff between the resolutionand the noise. Varying γ alters the solution for the weightsbetween a (local) pure least-squares solution and a minimumnoise solution. The value of γ must be subjectively selected to“optimize” the resulting image and depends on the measure-ment noise standard deviation ΔT and the chosen penaltyfunction. Following , we set the penalty function to aconstant J = 1 and the reference function to F = 1 over thepixel of interest, and F = 0 elsewhere.
TABLE ISSM/I CHANNEL CHARACTERISTICS . MOST FREQUENCIES HAVE
BOTH VERTICALLY POLARIZED (V) AND HORIZONTALLY
POLARIZED (H) RECEIVE CHANNELS
Previous investigators ,  have used BG to optimallydegrade the resolution of high-frequency channels to “match”that of lower frequency channels. Rather than do this, weoptimize the resolution of each channel independently to fixed-size grids.
In most previous applications of BG to spatial measurementinterpolation of radiometer data, the measurement layout andMRF were limited to small local areas and fixed geometriesto reduce computation and enable precomputation of the co-efficients , , . Azimuthally averaged antenna gainpatterns have also been used . Other investigators processed the measurements on a swath-based grid. The fixedgeometry yields only a relatively small number of possiblematrices, enabling computational saving by precomputation ofthe matrices. Unfortunately, the Earth-based grids used in theCETB produce a much more variable geometry than the fixedswath-based geometries, making this approach less viable sincesimilar shortcuts cannot be used.
For BG processing considered here, we follow  to define“nearby” as regions where the MRF is within 9 dB of the peakresponse. Outside this region, the MRF is treated as zero. Wecompute the solution separately for each output pixel usingthe particular measurement geometry antenna pattern at theswath location and Earth azimuth scan angle. This increasesthe computational load but results in the best quality images.The value of γ is subjectively selected for each channel but isheld constant for each of the rows in Table I.
We have found that the BG method occasionally producesartifacts due to poor condition numbers of the matrices that areinverted. To eliminate these artifacts, a median-threshold filteris used to examine a 3 × 3 pixel window area around each pixel.“Spikes” larger than 5 K above the local median are replacedwith the median value within the window. Smaller variationsare not altered.
The gridding and reconstruction methods can be appliedto a variety of radiometers such as SMMR, SSM/I, SSMIS,AMSR-E, and WindSat. However, to illustrate the methods andtheir tradeoff, in this paper, we concentrate on describing themethods as applied to a particular sensor, i.e., the SSM/I. Webriefly describe this sensor here. The same techniques can beapplied to the other sensors.
2770 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
Fig. 3. Illustration of the SSM/I swath. The antenna and feed are spun aboutthe vertical axis. Only part of the rotation is used for measuring brightnesstemperature: The rest is used for calibration. The incidence angle is essentiallyconstant as the antenna scans the surface. This diagram is for the aft-lookingF08 SSM/I. Later, SSM/Is looked forward but had the same swath width.
Fig. 4. Illustration of the individual 3 dB footprints of the various channelsshown at two different antenna rotation angles. Only footprints for the verticallypolarized channels are shown. Note the change in orientation (rotation) of thefootprints with respect to the underlying Earth-fixed grid reference .
The SSM/I is a total-power radiometer with seven operatingchannels (see Table I). An integrate-and-dump filter is usedto make radiometric brightness temperature measurements asthe antenna scans the ground track via antenna rotation .First launched in 1987, SSM/I instruments and their follow-ons,i.e., SSMIS, have flown on multiple spacecraft continuouslyuntil the present as part of the Defense Meteorological SatelliteProgram (DMSP) (F) satellite series.
The SSM/I swath and scanning concept is shown in Fig. 3.The antenna spin rate is 31.6 r/min with an along-track spacingof approximately 25 km. Multiple horns at 85 GHz providealong-track spacing for these channels at 12.5 km. The bright-ness temperature measurements are collected at a nominalincidence angle of approximately 53◦. A zoomed view ofthe arrangement of the antenna footprints on the surface fordifferent antenna azimuth angles is shown in Fig. 4.
V. RECONSTRUCTION PERFORMANCE SIMULATION
To compare the performance of the reconstruction tech-niques, it is helpful to use simulation. The results of thesesimulations inform the tradeoffs needed to select processingalgorithm parameters. A simple (but realistic) simulation ofthe SSM/I geometry and spatial response function is used togenerate simulated measurements of a synthetic Earth-centeredimage. From both noisy and noise-free measurements, non-enhanced (GRD), AVE, SIR, and BG images are created, witherror (mean and rms) determined for each case. This is repeatedseparately for each channel. The measurements are assumedto have a standard deviation of ΔT = 1 K. The results arerelatively insensitive to the value chosen.
Two different pass cases are considered: the single-pass caseand the case with two overlapping passes. Simulation showsthat the relative performance of SIR and BG are the same forboth cases; therefore, we show only one case in this paper. Sincemultiple passes are often combined in creating CETB images,the two-pass case is emphasized in the simulations presented.First, the size of the pixels for each channel must be determined.To generate images that can be embedded, the product pixelsize is restricted to fractional power of two of 25 km. That is,the pixel size Ps in kilometers is given by
where Ns is the pixel size scale factor, which is limited to thevalues 0, 1, 2, 3, and 4. The set of potential values of Ps arethus 25, 12.5, 6.25, 3.125, and 1.5625 km. For the simulation,the equal-area pixels are square.
An arbitrary “truth” image is generated with representativefeatures, including spots of varying sizes, edges, and areas ofconstant and gradient TB (see Fig. 5). The precise algorithmoptimums can depend on the truth image used . However,the other images considered in this paper produced similarresults; therefore, for clarity, the results from only a single truthimage are presented in this paper.
Based on the SSM/I measurement geometry, simulated loca-tions of antenna boresite at the center of the integration periodare plotted for a particular channel (37 GHz) in Fig. 6(a) for asingle pass. Images calculated in the polar regions can combinemeasurements from multiple passes of the spacecraft over thesame area. While the sampling for a single pass is fairly regular,the sampling from multiple overlapping passes tends to be lessregular. Fig. 6(b) illustrates the sampling resulting from twooverlapping passes. The variation in sample locations with each25-km grid element is apparent.
The MRF is modeled with a 2-D Gaussian function whose3 dB (half-power) point matches the footprint sizes given inTable I. The orientation of the ellipse varies over the swathaccording to the azimuth antenna angle. To apply the MRFin the processing, the MRF is positioned at the center of thenearest neighbor pixel to the measurement location and orientedwith the azimuth antenna angle. The values of the discrete MRFare computed at the center of each pixel in a box surroundingthe pixel center. The size of the box is defined to be the smallestenclosing box for which the sampled antenna pattern is larger
LONG AND BRODZIK: OPTIMUM IMAGE FORMATION FOR SPACEBORNE MICROWAVE RADIOMETER PRODUCTS 2771
Fig. 5. SSM/I 37-GHz simulation results in Kelvin. (Top Panel) The “true”brightness temperature image. (Left Column) Noise-free simulation results.(Right column) Noisy simulation results). (Top row) 25 km GRD. (Second row)AVE. (third row) SIR (20 iterations). (Bottom row) BG with γ = 0.85π. Errorstatistics are summarized in Table II.
Fig. 6. Illustrations of the measurement locations within a small area of theSSM/I coverage swath. (Top) Locations for a single orbit pass. (Bottom)Locations for two passes.
than a minimum gain threshold of −30 dB relative to the peakgain. A second threshold (typically −9 dB) defines the gaincutoff used in the SIR and BG processing. The latter thresholddefines the Nsize parameter used in .
The image pixel size defines how well the MRF can berepresented in the reconstruction processing and the simula-tion. Since we want to evaluate different pixel sizes in thissimulation, a representative plot of the MRF sampling for eachchannel for each pixel size under consideration is shown inFig. 7. Note that, as the pixel size is decreased, the sampledMRF more closely resembles the continuous MRF, therebyreducing quantization error; however, reducing the pixel sizeincreases the computation and size of the output products.
GRD images are created by collecting and averaging all mea-surements whose center falls within each 25-km grid element.For comparison with high-resolution BG and SIR images in thispaper, the GRD image is pixel-replicated to match the pixels ofthe SIR or BG images.
Separate images are created for both noisy and noise-freemeasurements. Error statistics (mean, standard deviation, andrms) are computed from the difference between the “truth” andestimated images for each algorithm option. The noise-only rmsstatistic is created by taking the square root of the difference ofthe squared noisy and noise-free rms values.
Both single and dual-pass cases were run as part of thispaper, but only the dual-pass cases are shown in this paper.Fig. 5 shows a typical simulation result. It shows the true image,and both noise-free and noisy images. The error statistics forthis case are given in Table II. For this pixel size, the imagesize is 448 × 224 with Ps = 3.125 km. In all cases, the erroris effectively zero mean. The nonenhanced results have thelarger errors, with the AVE results slightly less. The rmse isthe smallest for the SIR results. Visually, GRD and AVE aresimilar, although SIR images better define edges. The spots aremuch more visible in the SIR images than in the GRD images,although the SIR image has a higher apparent noise “texture.”BG results are discussed in a later section.
A. SIR Processing
Theoretically, SIR should be iterated to convergence to en-sure full signal reconstruction. This can require hundreds ofiterations . However, continued SIR iteration also tendsto amplify the noise in the measurements. By truncating theiteration, we can tradeoff signal reconstruction accuracy andnoise enhancement. Truncated iteration results in the signalbeing incompletely reconstructed (partial reconstruction) withless noise. The reconstruction error declines with further itera-tion as the noise increases.
To understand the tradeoff between number of iterations andsignal and noise, Fig. 8 shows noisy and noise-free SIR imagesfor several different iteration numbers (recall that AVE is thefirst iteration of SIR). Note that as the number of iterations isincreased, the edges are sharpened and the spots become moreevident. Fig. 9 plots the mean, standard deviation, and rmsversus iteration. Moreover, shown in this figure are the errorsfor the GRD and AVE (the first iteration of SIR) images. Thenoise texture grows with increasing iteration. We thus concludethat although iteration improves the signal, extended iterationcan overenhance the noise.
Plotting the signal reconstruction error versus noise powerenhancement as a function of iteration number in Fig. 9 can
2772 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
Fig. 7. Sampled MRF for different pixel sizes for an 85-GHz SSM/I channel. (Left) 6.25-km pixels. (Center) 3.125-km pixels. (Right) 1.5625-km pixels.
TABLE IIERROR STATISTICS FOR TWO-PASS SIMULATION FOR 37 GHz,
Ps = 3.125 km. SIR USES 20 ITERATIONS. BG USES γ = 0.85π
Fig. 8. SIR-reconstructed images for different iteration numbers in Kelvin.(Left column) Noise-free simulation results. (Right column) Noisy simulationresults. (Rows, top to bottom) 1 iteration (AVE), 10 iterations, 20 iterations,and 30 iterations.
help make a choice for the number of iterations that balancessignal and noise performance. Note that the GRD result is muchnoisier and that the signal error improves with each iteration ofSIR. Noting that we can stop the SIR iteration at any point,we somewhat arbitrarily choose a value of 20 iterations, whichprovides good signal performance and only slightly degraded
Fig. 9. (Top) SIR reconstruction error versus iteration number. (Upper left)mean error. (Upper right) rmse. The red line is the noisy measurement case,whereas the blue line is the noise-free measurement case. Green is the noisepower computed from the difference between the noisy and noise-free cases.The green line is vertically displaced for clarity. The large spot is the errorfor the GRD image, whereas the large square is for AVE. The “optimum”(minimum error) number of iterations occurs at the minimum of the red curve.For reference, the dashed vertical line is shown at 20 iterations. (Bottom) RMSnoise power versus rms signal error for each iteration, which extends from rightto left. The large spot denotes the GRD result, whereas the large square is theAVE result. The star is SIR at 20 iterations.
noise performance. This is the value used in Table II, where wesee that the overall error performance of the SIR reconstructionis better than the GRD result.
LONG AND BRODZIK: OPTIMUM IMAGE FORMATION FOR SPACEBORNE MICROWAVE RADIOMETER PRODUCTS 2773
TABLE IIITOTAL RMSE FOR NOISY TWO-PASS SIMULATION USING
Ns = 3 (Ps = 3.125) AND 20 SIR ITERATIONS
This analysis is repeated for different values of Ns, i.e.,the number of passes, and for each radiometer channel. Whilethe numerical values of the rmse change, the overall rankingand relative spacing of the GRD and SIR values are the samefor all cases. Based on these detailed results (not shown), thefollowing observations can be made.
1) With proper choice of the number of iterations SIRalways has better signal performance (as well as betterspatial resolution) than GRD.
2) Iterating SIR too long causes it to have worse noiseperformance than GRD, although the signal performanceimproves for longer iterations.
3) Based on the rmse comparison, SIR performance isslightly better than BG with the optimum γ.
In general, we want to use a small Ns to minimize com-putation and to minimize the number of iterations. Based onNyquist criteria for sampling the response pattern, Ns = 2 isthe minimum usable value. With the idea that we want to keepthe same values for all channels, if possible, for consistency,it appears that Ns = 3 (i.e., Ps = 3.125 km) provides the bestoverall performance for all channels and that 5–20 iterationsprovide a reasonable tradeoff between signal and noise. UsingNs = 3, Table III provides a performance comparison for thermses of GRD, AVE, and SIR for the different channels using20 iterations. Fig. 5 compares the resulting noisy simulationresults. Note that, although the pixel size is 3.125 km, theeffective spatial resolution of the images is, of course, coarserthan this. Recall that at least some of the extra pixel resolutionis required to properly process the signal to meet the Nyquistsignal representation requirements and represent the higherfrequency content of the high-resolution images. It should benoted that the precise minimum value depends on the exactnoise realization; however, the optimum can only be computedin simulation; therefore, we have chosen the nominal optimumand use it for the processing of real data.
In summary, SIR provides better spatial resolution and loweroverall rmse than conventionally gridded (GRD) products. Thereconstruction does tend to enhance the noise, but this is offsetby reduced signal error. The total error can be controlled by thenumber of iterations to tradeoff noise and resolution.
B. BG Processing
The BG approach requires selection of a subjective tuningparameter. It also requires significantly more computation thandoes SIR. Previous investigators who worked on swath-basedgrids were able to coarsely quantize the possible measurementpositions to reduce the number of matrix inversions required.This enabled them to generate precomputed approximate
solutions , . However, as noted previously, the situationis different when working with Earth-based grids. Note that,Fig. 4, the measurement centers are irregularly arranged withrespect to the Earth-located pixel grid. This limits any abilityfor using similar preprocessing techniques to speed the BGcomputation. Further, to avoid the approximations used withfixed geometries, we use the actual measurement positions anda general pixel grid so that reconstruction can be done onthe Earth-located pixel grid . Another advantage of thisapproach is that all data from overlapping swaths (subject toltod) can be used, whereas previous implementations used datafrom one swath for any given Earth grid cell , . Thegeneral form we use requires creating and inverting a matrixfor each image pixel.
In the BG simulations in the following, the same simulatedSSM/I measurements are used as in the SIR simulations. In, note that SIR and BG results appear similar, but SIR hasslightly better performance in simulation and is much fasterthan BG. Our simulations for SSM/I confirm this conclusion.As noted, the BG γ controls the regularization and relativeweighting between signal reconstruction and noise enhance-ment. The value of γ can range from 0 to π/2. Note that, forsimplicity in the captions and plots, the symbols γ ′ or g aresometimes used, which are related to γ by γ = (π/2)γ ′ andγ = πg.
A BG image for a particular γ closely resembles a SIR imagefor a particular iteration number. For small values of γ ′, thenoise is the most enhanced but the features are sharpest. Forlarger values of γ ′, the noise texturing is reduced but featuresare smoothed. A plot of the rmse versus γ ′ is shown in Fig. 10.Note that noise-free and noisy results are shown both for BGand BG after median filtering. Due to a poorly conditionedmatrix, some BG-estimated pixels have extreme values. Thesecan be suppressed by applying a 3× 3 median filter after the BGprocessing. The median filter can significantly reduce the rmsnoise and artifacts in the image without significantly degradingthe image quality. The median filter is edge preserving and sohas minimal effect on the image quality.
Similar to the analysis of number of iterations for SIR, itis useful to compute the noise and signal rmse, which varieswith the value of γ ′. An example for the 37-GHz channel withNs = 3 is shown in Fig. 10. A numerical comparison of theresults is shown in Table II. BG for the optimum γ is noisierthan SIR with the optimum number of iterations.
Other examples of BG images versus different gammaparameters were considered. Large values of Ns result inexcessive (and impractical) run times. Fortunately, the rmsevaries only slightly with changes in Ns and the same valueselected for SIR processing can be used. All cases have aminimum total rmse near g = 0.85; therefore, for consistency,we adopt this value, i.e., γ = 0.85π, for all channels and allvalues of Ns.
We find that BG requires at least an order of magnitude morecomputation time than SIR. For larger values of Ns, it requiresseveral orders of magnitude more computation time; hence, thedesire for small values ofNs. The choice of γ does not affect thecomputation, but changing the antenna gain cutoff from −9 dBto larger values can reduce the number of local measurements
2774 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
Fig. 10. (Top) BG error versus versus γ′. (upper left) mean error. (Upper right)rmse. The location of the optimum (i.e., the minimum rmse) values is indicatedwith asterisks. (Bottom) RMS noise versus RMS signal error for different γ′.The solid line is noisy BG, whereas the dotted line is after thresholded medianfiltering. These lines often coincide in the plots.
included in the matrix inversion and thus the required CPUtime. Changing the gain threshold is not considered here.
In summary, BG and SIR provide similar results, and bothprovide better spatial resolution and overall rmse than conven-tionally gridded (GRD) products. BG is, however, computation-ally demanding. Final selection of BG versus SIR for CETBwill rely on input from the user community.
VI. RECONSTRUCTION SENSITIVITY TO INACCURACY
IN THE DESCRIPTION OF THE MRF
As previously noted, the MRF for some sensors is not knownvery well. Even for those for which the MRF is known, there areuncertainties (errors) in the description of the MRF. This leadsto the following question: How sensitive is the reconstructionto the accuracy of the MRF? In pursuing this question, weconsider only the SIR algorithm in this paper. However, we findsimilar performance using BG. We note that we are interested
Fig. 11. Plots of the noisy total rmse resulting from simulating measurementswith the MRF and using an erroneous MRF from (20) versus fractional powerρ. The dotted curve is the SIR error at 20 iterations. The solid curve shows thermse resulting from minimizing the total rmse versus SIR iteration number.
in the partial reconstruction case when only a relatively small(5–20) number of SIR iterations is performed.
To address the question of reconstruction sensitivity, anexperimental study is conducted in which simulated measure-ments of a synthetic scene are generated using the full MRFpreviously described. Then, different (erroneous) MRF descrip-tions are used in the reconstruction process. The results fromthe correct description and the erroneous descriptions are thencompared.
Although other erroneous MRFs were considered in a largerstudy , in this paper, we consider the family of erroneousMRFs defined to be a power of the true MRF. That is, theMRF used for reconstruction R′(x, y) is computed from thetrue MRF R(x, y) using
R′(x, y) = Rρ(x, y) (20)
where 0 < ρ < 3. As ρ is varied in the range of 0.25–3, thearea of the 3-dB footprint changes, and the response patternrolloff characteristics change. Fig. 11 plots the total noisy rmseversus ρ for this case. Two curves are shown. One is for afixed number of iterations (20 in this case), and the other isthe rmse resulting when selecting the number of SIR iterationsthat minimize the total rmse. This is the optimum number ofSIR iterations. Note that, in all cases, the variation in the rmseis small, and the difference between the fixed and the optimumnumber of iterations is also very small.
These simulation results reveal that using the correct MRFfor reconstruction minimizes the error, but modest distortionsin the MRF used in the reconstruction have a limited impact onthe accuracy of the reconstruction results. The variation in totalrmse with MRF distortion is small for all channels and cases.Thus, the results of the reconstruction are not particularly sen-sitive to the accuracy of the MRF, and we can successfully useapproximate MRF models. This is a fortunate result because it
LONG AND BRODZIK: OPTIMUM IMAGE FORMATION FOR SPACEBORNE MICROWAVE RADIOMETER PRODUCTS 2775
means that precise antenna pattern descriptions are not requiredin computing MRF.
To explain this result, consider that since the noise is ampli-fied as the signal is enhanced, there is a tradeoff between thesignal reconstruction error and the noise increase. This tradeoffleads us to truncate the iterative reconstruction process beforeit is complete, i.e., the result is only a partial reconstruction.The simulations show that the partial reconstruction can toleratemodest errors in the MRF description and still yield reasonableestimates of the desired signal. Not shown is that when theerroneous MRF is used to attempt to fully reconstruct thesignal, the final signal can be significantly distorted comparedto the signal resulting from the correct MRF description.
VII. ACTUAL DATA
While the previous results are based on simulation, here,we compare GRD and enhanced-resolution products using ac-tual data (see Fig. 12). These evening ltod images are small(250 km × 250 km) subimages extracted from the full EASE-Grid-2.0 Northern Hemisphere grid for day 61 of 1997 usingthe selected grid size and algorithm parameters derived fromsimulation. A visual comparison of the images reveals im-proved detail in the SIR and BG images compared with theGRD images. Note that, when using actual data, the true TB
values are not known; therefore, errors cannot be computed.As expected, the GRD images are blocky, whereas the high-
resolution images exhibit finer resolution. Subjectively, the SIRimages have the highest contrast and appear more detailedthan the BG images, but there is also somewhat more noise inthe SIR images compared with the BG images. Note that theeffective resolution varies between channels, with the highestfrequency channels (which have the smallest MRFs) providingthe finest resolution. The differences in TB with polarizationshow that vertically polarized images appear in general brighterthan horizontally polarized images. Further interpretation of theTB images is left to other papers.
The NASA-sponsored Calibrated Passive Microwave DailyEASE-Grid 2.0 Brightness Temperature ESDR (CETB) projectis producing a multisensor multidecadal time series of griddedradiometer products from consistently calibrated TB measure-ments from SMMR, SSM/I, SSMIS, and AMSR-E. The CETBproduct includes both conventional 25-km gridded images andhigh-resolution radiometer products designed to support cli-mate studies. This paper has considered two primary high-resolution image reconstruction algorithms: the BG approachand the radiometer form of SIR algorithm. In effect, for eachpixel, BG inverts a local matrix that describes the interaction ofmeasurements near the pixel of interest and a different matrixthat depends on the relative location and spatial measurementresponse pattern of the measurements. This matrix is regular-ized by including the noise covariance. The iterative SIR algo-rithm reconstructs the image from the measurements using themeasurement locations and response patterns. SIR is equivalentto inverting the full matrix reconstruction matrix for the entire
Fig. 12. Subimages extracted from a one-day CETB Northern HemisphereSSM/I EASE-Grid 2.0 image product: (left column) GRD, (Center Column)SIR, and (Right Column) BG, and (each row, top to bottom) for channels 19H,19V, 37H, 37V, 85H, and 85V. The area shown spans 250 km × 250 km and iscentered over Baffin Island west of Greenland, which is partially visible in thelower right corner. No-data pixels in the GRD images have been filled with themedian of nearby pixels units in Kelvin.
image but is regularized by truncating the iteration, resulting inonly partial reconstruction. Both SIR and BG provide higherspatial resolution surface brightness temperature images withsmaller total error compared with conventional DIB griddedimage formation.
A tradeoff study is summarized for selecting optimum gridresolution, the number of SIR iterations, and the BG gammaparameter. Although the optimum values vary somewhat be-tween sensors and sensor channels, we constrain the selectedparameters to a common set of values. We found that althoughboth image formation approaches are effective in improvingthe spatial resolution of the surface brightness temperatureestimates, SIR however requires less computation. Our analysis
2776 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 5, MAY 2016
TABLE IVGRIDS AND RECONSTRUCTION PARAMETERS PLANNED FOR CETB SENSORS AND CHANNELS.NOT ALL CHANNELS ON EACH SENSOR ARE PROCESSED. GRIDS ARE NESTED TO OVERLAY
of the sensitivity of the reconstruction to the accuracy of theMRF shows that the partial reconstruction selected based on thenoise tradeoff can tolerate errors in the description of the sensormeasurement response function. This simplifies the processingof sensor data for which the measurement response function isnot known.
A summary of the reconstruction parameters selected forthe CETB project for each sensor channel is summarized inTable IV.
Here, we consider the spatial frequency information withinan SSM/I measurement by computing the wavenumber spec-trum of the MRF. Fig. 13 provides a conceptual illustrationof the projection of the sensor antenna pattern on the Earth’ssurface in the elevation plane and the geometry definitions usedin the following. From the geometry and temporarily assuminga spherical Earth, the nominal slant range R between the sensorand boresite location can be computed from the spacecraftheight H , the incidence angle θi, and the (local) radius of theEarth Re using
[θi − sin−1 Re sin(180− θi)
Aligning the x coordinate with the look direction at azimuthangle φ = 0, on a locally tangent plane, the approximate xdisplacement is dx ≈ Rdθi/ sin θi. Since the incidence anglefor the SSM/I is approximately 53◦, 1/ sin θi ≈ 1.252; there-fore, the nominally circular antenna pattern is elongated onthe surface in the range direction by about 25%, resultingin an elliptical footprint on the surface. As previously noted,the instantaneous antenna pattern is smeared in the rotation(azimuth) direction by the temporal signal averaging.
Given the elevation angle and height, for a particular dis-placement Δx along the Earth’s surface from the intersectionof the antenna boresite vector at elevation angle, the antennaelevation angle offset Δξ is computed by first determining theEarth angle α0 (see Fig. 13) using
ξ = sin−1
(Re +H)sin(180− θi)
α0 = θi − ξ −Δx/Re. (23)
Fig. 13. (Top) Conceptual diagram illustrating the projection of the instanta-neous radiometer antenna pattern on the Earth’s surface. (Bottom) Geometryfor computation.
The antenna elevation offset angle corresponding to the pointof interest at Δx is computed as
R0 =√(Re +H)2 +R2
e − 2(Re +H)Re cosα0 (24)
Δξ = ξ − sin−1
These equations provide a formula to relate surface displace-ment from the boresite to the change in the elevation angle.
LONG AND BRODZIK: OPTIMUM IMAGE FORMATION FOR SPACEBORNE MICROWAVE RADIOMETER PRODUCTS 2777
It is known that the far-field antenna pattern can be expressedas the Fourier transform of the electric field across the effectiveaperture of the antenna . Since the effective aperture is, bydefinition, finite extent, this implies that the antenna patternis angularly bandlimited. This argument has been used tosuggest that the wavenumber spectrum (the 2-D spatial Fouriertransform) of the MRF is similarly bandlimited. However, itshould be noted that the aperture-to-far-field Fourier transformis computed in angular units, whereas the spectrum of the MRFis in km−1 on the surface. As derived earlier, there is a nonlineartransformation between angle and surface distance. Further, theangular Fourier transform is periodic in angle, whereas the sur-face spectrum Fourier is computed over a finite domain and istherefore infinite in extent. In the following, we consider theseissues in more detail and compute the approximate spectrum ofthe MRF.
As previously described, the MRF results from projecting theantenna pattern onto the surface and integrating the movingpattern over the integration period. The projection stretchesand compresses the pattern on the Earth’s surface. Due to thecurvature of the Earth, not all of the antenna pattern is projectedonto the Earth’s surface. Thus, the MRF contains only part ofthe antenna pattern, i.e., the pattern is clipped. This can bemodeled as the multiplication of the stretched antenna patternby a rectangular window or boxcar function. The inverseR termin the spherical propagation factor has the effect of taperingthe projection of the antenna gain onto the surface MRF. Forsimplicity, this factor is ignored in the following discussion. In-tegration has only a small effect on the projected elevation gain.
As evident from (22)–(25), the projection in elevation isnonlinear. The MRF on the surface is a nonlinearly stretchedand windowed (clipped) version of the original antenna pattern.In principle, the wavenumber (spatial) spectrum of the MRFcan be analytically derived from the projecting and clippingfunctions. However, this is quite complicated, and we resortto numerical methods. Recall from signal processing theorythat the Fourier transform of the product of two functions isthe convolution of the Fourier transforms of the individualfunctions. In this case, the convolution yields a wider regionof support in frequency domain than either of the individualfunctions. Furthermore, since the clipping window function isfinite length, its Fourier transform is infinite in extent (though itcan have nulls). The convolution of an infinite-extent functionwith any function results in an infinite-extent function. It thusbecomes apparent that the region of support of the MRF is notbounded, i.e., at the surface, the MRF is not truly bandlimited,although the aperture illumination pattern is.
The wavenumber (spatial frequency) of the MRF informsus about the information content in the radiometer measure-ments. The nonlinearity in the projection from antenna patternto the MRF produces a broader region of support for MRFwavenumber spectrum than suggested from the finite region ofsupport of the antenna pattern illumination function, althoughthere is attenuation of the high-wavenumber portions of thespectrum and portions of the high-wavenumber spectrum areunrecoverable due to low gain.
The top two panels of Fig. 14 illustrate a slice through theidealized 19-GHz SSM/I pattern over the full angular extent of
Fig. 14. (Top panel) Magnitude squared plot of elevation slice through anidealized 19-GHz SSM/I channel antenna pattern. (Second panel) MagnitudeFourier transform. This is the magnitude of the aperture illumination function.(Third panel) Antenna pattern projected onto the Earth versus displacementalong the surface. The x-axis is the range over which the platform is visiblefrom the surface. Note the variable spacing of the sidelobes. (Bottom panel)Magnitude Fourier transform of the slice of the projected antenna pattern.The vertical line is at −20 dB and corresponds to a 27 km spatial scale. Forcomparison, the 3-dB scale corresponds to 43 km.
the pattern (the effects of spacecraft blockage are ignored), andthe corresponding Fourier transform, which is Ea(ra), has afinite region of support as expected. The other channels havesimilar results, although for shorter wavelengths, the patternis narrower and the frequency support is wider. Projecting theelevation-slice antenna pattern on to the Earth’s surface usingtypical SSM/I measurement geometry results in the gain patternon the Earth’s surface, as shown in the bottom panels of Fig. 14.The wavenumber spectrum has units of inverse ground distance,with a scale distance that is twice the inverse of the wavenum-ber. The region of support in surface wavenumber has a taperedrolloff. As expected, broader region of support confirms that wecan recover as least somewhat more spatial information thanmerely assuming the spatial scale is set by the 3 dB footprint.
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David G. Long (S’80–SM’98–F’08) received thePh.D. degree in electrical engineering from the Uni-versity of Southern California, Los Angeles, CA,USA, in 1989.
From 1983 to 1990, he worked for NASA JetPropulsion Laboratory (JPL), where he developedadvanced radar remote sensing systems. While atJPL, he was the Project Engineer on the NASA Scat-terometer (NSCAT) project, which flew from 1996to 1997. He also managed the SCANSCAT project,i.e., the precursor to SeaWinds, which was flown in
1999 on QuikSCAT, in 2002 on ADEOS-II, and in 2014 on the InternationalSpace Station. He is currently a Professor with the Department of Electrical andComputer Engineering, Brigham Young University (BYU), Provo, UT, USA,where he teaches the upper division and graduate courses in communications,microwave remote sensing, radar, and signal processing and is the Directorof the BYU Center for Remote Sensing. He is the principal investigator onseveral NASA-sponsored research projects in remote sensing. He has over400 publications in various areas, including signal processing, radar scatterom-etry, and synthetic aperture radar. His research interests include microwaveremote sensing, radar theory, space-based sensing, estimation theory, signalprocessing, and mesoscale atmospheric dynamics.
Dr. Long serves as an Associate Editor for IEEE GEOSCIENCE AND RE-MOTE SENSING LETTERS. He received the NASA Certificate of Recognitionseveral times.
Mary J. Brodzik received the B.A. (summa cumlaude) degree in mathematics from Fordham Univer-sity, New York, NY, USA, in 1987.
Since 1993, she has been with the National Snowand Ice Data Center (NSIDC) and with the Coop-erative Institute for Research in Environmental Sci-ences, University of Colorado, Boulder, CO, USA,where she is currently a Senior Associate Scientist.Since coming to NSIDC, she has implemented soft-ware systems to design, produce, and analyze snowand ice data products from satellite-based visible
and passive microwave imagery. She has contributed to the NSIDC datamanagement and software development teams for the NASA Cold LandsProcesses Experiment and Operation IceBridge. As a Principal Investigator onthe NASA Making Earth Science Data Records for Use in Research Environ-ments (MEaSUREs) Calibrated Enhanced-Resolution Brightness Temperatureproject, she is managing the operational production of a 37-year EASE-Grid2.0 Earth Science Data Record of satellite passive microwave data. She is usingMODIS data products to derive the first systematically-derived global mapof the world’s glaciers. For the U.S. Agency for International Development,she is developing snow and glacier ice melt models to better understand thecontribution of glacier ice melt to major rivers with headwaters in High Asia.Her experience includes software development, validation and verificationon Defense Department satellite command and control, and satellite trackingsystems. Her research interests include optical and passive microwave sensingof snow, remote sensing data gridding schemes, and effective ways to visualizescience data.